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Workbook 05-Lab 4- Hypothesis Testing for a Population Proportion Author: Brenda Gunderson, Ph.D., 2015 License: Unless otherwise noted, this material is made available under the terms of the Creative Commons Attribution- NonCommercial-Share Alike 3.0 Unported License: http://creativecommons.org/licenses/by-nc-sa/3.0/ The University of Michigan Open.Michigan initiative has reviewed this material in accordance with U.S. Copyright Law and have tried to maximize your ability to use, share, and adapt it. The attribution key provides information about how you may share and adapt this material. Copyright holders of content included in this material should contact [email protected] with any questions, corrections, or clarification regarding the use of content. For more information about how to attribute these materials visit: http://open.umich.edu/education/about/terms-of-use. Some materials are used with permission from the copyright holders. 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Our determination DOES NOT mean that all uses of this third-party content are Fair Uses and we DO NOT guarantee that your use of the content is Fair. To use this content you should conduct your own independent analysis to determine whether or not your use will be Fair. Lab 4: Hypothesis Testing for a Population Proportion Objective: In this lab, you will learn an important statistical technique that will allow you to answer the question, “Was our observation due to chance, or is it more significant?” The objective is to guide you through the ideas behind tests of statistical significance and the statistical language involved. This lab first presents a general overview of testing. Then, the In-Lab Project discusses the large sample Z-test for a population proportion, as well as providing practice in answering the question of whether an event can be attributed to chance. Application: Lucy is an Ann Arbor resident who is considering the possibility of running for a seat on the City Council. To decide if she should invest time and money in her campaign, she wants to conduct some research to see if there is evidence that she could win the election (i.e. she could receive a majority of all the votes). Overview: Hypothesis Tests: A test of hypotheses (or significance test) is a procedure designed to assess what the evidence provided by the data says about some statement about a population parameter. When we conduct this hypothesis test for Lucy, we are interested in testing if the population proportion of votes that would be cast for Lucy is greater than 0.5, ensuring her victory. Hypothesis Testing Steps: 1. Determine appropriate null and alternative hypotheses and set ∝. 2. Check assumptions for performing the test. 3. Calculate the test statistic and determine the p-value. 4. Evaluate the p-value and report a conclusion in the context of the problem. The first step of this hypothesis test is to identify the hypotheses; this step is crucial, as it dictates the procedures for the remainder of the test. The null hypothesis, H0, represents the status quo or statement of no effect. The alternative hypothesis, Ha (or sometimes H1), represents the experimenter's new model, or what the experimenter would like to show. For our application regarding Lucy, we could define p to represent the true proportion of all Ann Arbor residents who would vote for Lucy. We want to test the hypothesis H0: p = 0.5 against the alternative Ha: p > 0.5. Both of the hypotheses are postulated about the same population proportion. The alternative hypothesis can take three different forms – it may be a denial of the null hypothesis (uses≠; called a two-sided test), or it may specify a direction of interest (uses > or <; called a one-sided test). Notice that we never test for equality in Ha. The purpose of a significance test is to assess whether or not the observed data are consistent with the null hypothesis (within the reasonable bounds of sampling variability). If we collect our sample and compute a population proportion of votes for Lucy that is much higher than the 0.5 we specified in the null hypothesis, we have evidence to reject that null hypothesis. To help us make this decision, we use a test statistic, which represents a summary of the data. When we conduct a test about a population proportion, we are performing a Large Sample Z test. The test statistic for this Large sample z test has the following form: �−� � = 0 . �0 (1−�0)/� This �! is the value specified in the null hypothesis. For Lucy’s test: �! = 0.5. The sample proportion of people who will vote for Lucy is our �. The test statistic tells us how many standard errors the sample proportion, � , is from the test value, p0. The test relies on two key assumptions: that we have a random sample (our sample of responses is representative of the large population of all such responses), and that our sample size is large enough. To achieve a large enough sample size, we need at least 10 hypothesized yes answers and at least 10 hypothesized no answers, or said another way, under the null hypothesis, would we expect to see at least 10 yes answers and at least 10 no answers. Conditions: ��!>10 and �(1 − �!)>10 The Z test statistic has a known probability distribution (under the null hypothesis), and will be examined for evidence in favor of or against H0. Under the null hypothesis that the true proportion of votes for Lucy is equal to 0.5, the test statistic Z has a standard normal distribution. In hypothesis testing, another frequently reported value is the p-value, a number that is used to indicate the degree of significance of the data. The p-value is the probability of getting a test statistic as extreme or more extreme than the observed value of the test statistic, assuming the null hypothesis is true. For Lucy’s test, the p-value is the probability of getting a z test statistic like we did or greater, assuming that the true proportion of votes for Lucy is equal to 0.5. We must decide in advance how much evidence against H0 we will require for rejection. This designated amount of evidence is called the level of significance, denoted by α (alpha). Common values of α are 0.01, 0.05, and 0.10. If the p-value is less than or equal to α, we make the decision to reject H0. For Lucy’s test, if we encounter a p-value smaller than the significance level, we would decide to reject the hypothesis that the true proportion of votes for Lucy is exactly 0.5. If we reject the null hypothesis, the results of the test are said to be statistically significant at level α. A “significant” result in the statistical sense does not necessarily imply an “important” result in the practical sense. It simply means that such a difference from the null hypothesis is not likely to happen just by chance. Of course, no procedure is perfect, and as such, there are two types of errors possible during hypothesis testing. If the null hypothesis is true but the decision is to reject H0, then we say that a Type I error has occurred. If we commit a Type I error for Lucy’s test, we would reach the decision that Lucy will win the election by receiving more than 50% of the votes, when in fact the true proportion of votes cast for Lucy is 0.5. Lucy would invest her time and money in her campaign when in fact she will not win the election. A Type II error occurs when the alternative hypothesis is true, but we fail to reject H0. A Type II error could be committed in Lucy’s test if we decide Lucy will lose the election by receiving 50% of the votes or less, when in fact Lucy will receive greater than 50% of the votes and win the election. Lucy would decide not to run in the election when she would have won a seat on the City Council. Each type of error has a probability of occurring. If the null hypothesis is true, the level of significance, α, is also the probability of a Type I error, while the probability of a Type II error is denoted by β. Truth Decision Made Result Associated Probability Reject H Type I Error α 0 H0 True Do Not Reject H0 Correct Decision 1 – α Reject H0 Correct Decision 1 – β = power Ha True Do Not Reject H0 Type II Error β Another important component of a hypothesis test is its power. The power of a test measures its ability to detect an alternative hypothesis when it is true. Power of a particular test is calculated as the probability that the test will reject H0 when the alternative hypothesis is true.
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